Geoscience Reference
In-Depth Information
on an intermediate timescale too: the pole wanders by a few degrees a century.
This wandering has been measured throughout historical time and is termed
secular variation
.However, it appears that, on average over geological time,
the geomagnetic field axis has been aligned along the Earth's spin axis (i.e.,
on average the geomagnetic poles have been coincident with the geographic
poles). This means that, to a first approximation, the geomagnetic field can be
modelled as the field of a dipole aligned along the geographic north-south axis.
This assumption is critical to all palaeomagnetic work: if the dipole axis had
wandered randomly in the past and had not, on average, been aligned along the
geographic axis, all palaeomagnetic estimates of past positions of rock samples
would be meaningless because they would be relative only to the position of the
geomagnetic pole at the time each sample acquired its permanent magnetization
and would have nothing at all to do with the geographic pole.
The
magnetic potential
from which the Earth's magnetic field is derived can
be expressed as an infinite series of spherical harmonic functions. The first term
in this series is the potential due to a dipole situated at the centre of the Earth. At
any position
r
, from a dipole, the magnetic potential
V
(
r
)isgivenby
1
4
π
r
3
m
·
r
V
(
r
)
=
(3.1)
where
m
is the dipole moment, a vector aligned along the dipole axis. For the
Earth
m
is 7.94
10
22
Am
2
in magnitude. The magnetic field
B
(
r
)atany position
r
can then be determined by differentiating the magnetic potential:
×
B
(
r
)
=−
µ
0
∇
V
(
r
)
(3.2)
10
−
7
kgmA
−
2
s
−
2
is the magnetic permeability of free space
(A is the abbreviation for amp).
To apply Eq. (3.2)tothe Earth, we find it most convenient to work in spherical
polar coordinates (
r
,
where
µ
0
=
4π
×
the longitude
or azimuth on the sphere, as shown in Fig. A1.4). The magnetic field
B
(
r
)is
then written as
B
(
r
)
θ
,
φ
), (
r
is the radius,
θ
the colatitude and
φ
(
B
r
,
B
θ
,
B
φ
)inthis coordinate system.
B
r
is the radial
component of the field,
B
θ
is the southerly component and
B
φ
is the easterly
component. (See Appendix 1 for details of this and other coordinate systems.)
In spherical polar coordinates, if we assume that
m
is aligned along the
negative
z
axis (see caption for Fig. 3.2), Eq. (3.1)is
=
1
4
π
r
3
m
·
r
V
(
r
)
=
mr
cos
θ
4
π
r
3
=−
m
cos
θ
4
π
r
2
=−
(3.3)
Substitution of Eq. (3.3) into Eq. (3.2)gives the three components (
B
r
,
B
θ
,
B
φ
)of
the magnetic field due to a dipole at the centre of the Earth. The radial component
